Problem: Simplify the following expression and state the condition under which the simplification is valid: $z = \dfrac{r^2 + 10r + 16}{r^2 - r - 72}$
Explanation: First factor the expressions in the numerator and denominator. $ \dfrac{r^2 + 10r + 16}{r^2 - r - 72} = \dfrac{(r + 2)(r + 8)}{(r - 9)(r + 8)} $ Notice that the term $(r + 8)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(r + 8)$ gives: $z = \dfrac{r + 2}{r - 9}$ Since we divided by $(r + 8)$, $r \neq -8$. $z = \dfrac{r + 2}{r - 9}; \space r \neq -8$